5
$\begingroup$

This is cross-posted from Math.SE at the recommendation of a commenter.

I'm reading M. Rosenlicht's 1956 paper, "Some Basic Theorems on Algebraic Groups" [link], and having trouble with some of the language and notation. In particular, I have two questions:

1) Rosenlicht defines an algebraic group as a variety with a group structure on the points. He pointedly does not assume the ground field $k$ is algebraically closed. As he was writing prior to Grothendieck's development of scheme theory, I am not sure how to interpret "points". In this context, does he mean the $k$-points? The $\overline k$-points i.e. the geometric points? Or what?

2) With $G$ an algebraic group, and $V$ a variety with an action of $G$, all defined over a field $k$, Rosenlicht frequently writes expressions like "$k(g,v)$" with $g\in G, v\in V$; for example, on p. 403, we have

We say $G$ operates on $V$ (or that $V$ is a pre-transformation space for $G$) if for each component $G_\alpha$ of $G$ we are given a rational map $g\times v\to g(v)$ of $G_\alpha\times V\rightarrow V$ such that if $k$ is a field of definition for $G$, $V$, and each of these rational maps and if $g_1\times g_2 \times v$ is a generic point over $k$ of $G_\alpha \times G_\beta \times V$ ($G_\alpha$, $G_\beta$ being any components of $V$) then

$$(1)\:\:\:\: g_1(g_2(v)) = g_1g_2(v).$$ $$(2)\:\:\:\: k(g_1,g_1(v)) = k(g_1,v).$$

What's meant by $k(g_1,v)$ here? I want it to be a residue field but it has 2 arguments. Is it the composite of the two residue fields? (In which case, the answer to question 1 cannot be "$k$-points"?)

Aside: I am particularly interested in Theorem 2 on p. 407 ("Rosenlicht's theorem"). However, my goal is to be able to read Rosenlicht's paper, so I would prefer a gloss of his usages over a reference to a more modern statement and proof.

Addendum: The accepted answer contains a link to another question/answer which links to this very helpful piece by Raynaud in the Sept. 1999 Notices, discussing Weil's foundational approach (which turns out to be the context for Rosenlicht's paper). I include the link here for the sake of self-containedness of the question/answer pair.

$\endgroup$
11
$\begingroup$

Weil's foundations (which Rosenlicht uses) use a universal domain, which is an algebraically closed extension of $k$ of infinite transcendence degree. There is a discussion of universal domains here. In this language, "point" means a point with coordinates in a universal domain. The expression $k(g,v)$ means the subfield of the universal domain generated over $k$ by the coordinates of the points $g$ and $v$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.